Network optimization problems : algorithms, applications, and complexity

NETWORK PROBLEMS

SERIES ON APPLIED MATHEMATICS Editor-in-Chief: Frank Hwang Associate Editors-in-Chief: Zhong-ci Shi and Kunio Tanabe

Vol. 1

International Conference on Scientific Computation ed. T. Chan and Z.-C. Shi

P"NETWORK PROBLEMS ALGORITHMS, APPLICATIONS AND COMPLEXITY Editors

Ding-Zhu Du Department of Computer Science University of Minnesota and Institute of Applied Mathematics Academia Sinica, Beijing

P a n o s M. P a r d a l o s Department

of Industrial and Systems University of Florida

Engineering

Y J * World Scientific wfc

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farcer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Library of Congress Cataloging-in-Publication Data Network optimization problems : algorithms, applications, and complexity / editors, Ding-Zhu Du, Panos M. Pardalos. p. cm. — (Series on applied mathematics; v. 2) Includes bibliographical references. ISBN 9810212771 1. System analysis. 2. System design. 3. Mathematical optimization. I. Du, Dingzhu. II. Pardalos, P. M. (Panos M.), 1954III. Series. T57.6.N47 1993 OO3-dc20 93-16337 CIP

Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orby any means, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.

Printed in Singapore by Utopia Press.

V

Preface T h e field of networks is a lively one, both in terms of theoretical developments and in t e r m s of t h e diversity of its applications. Many problems of design and analysis of large systems can be formulated and solved using techniques of network theory. Such problems include communication systems, electrical networks, computer networks, transportation, scheduling of industrial processes, facility location, and modeling of combinatorial optimization problems. Network theory originated m a n y years ago, before our information age. In the eighteenth century, Euler solved the famous Konigsberg Bridge problem and later Kirchoff initiated the theory of electrical networks. But it was not until late last century, when Bell invented t h e telephone, t h a t many areas of network theory were stimulated. After t h e appearance of t h e first graph theory book (by D. Konig) in 1936, there was tremendous development regarding t h e theory and applications of networks. Hitchcock proposed t h e first complete algorithm for t h e transportation problem in 1941, Dantzig proposed t h e simplex algorithm for linear programming in 1947, and algorithms for t h e m i n i m u m spanning tree (Kruskal, 1956) and shortest p a t h problems were proposed ( P r i m , 1957). During t h e same period, the first commercial computers became available. As it happened with many other areas of research, the fields of computer science and networks influenced each other in many respects. In 1962 the book by Ford and Fulkerson on "Flows in Networks" appeared. W i t h t h e development of new d a t a structure techniques and t h e theory of computational complexity we entered a new era of algorithmic developments in networks. During t h e second half of our century we saw major technological developments in all areas of h u m a n endeavor and particularly in information processing. C o m p u t e r networks play a vital role in providing fast, reliable, cost-effective means of communication and information sharing. In addition, network techniques and computer technology enable us to solve large-scale network models t h a t appear in applications such as transportation and telecommunications. It is clear t h a t the theory and applications of networks is so great that this book could not give a full account and systematic t r e a t m e n t of the subject in its entirety. It is our intention to introduce a number of special topics in order to show t h e spectrum of recent research activities and the richness of ideas in the development of algorithms and t h e applications of networks. While we were able to provide only a glimpse of this expansive field, we felt t h a t this glimpse would allow the reader to sense t h e breadth and t h e depth of t h e field. We would like to take the opportunity to thank t h e authors of the papers, t h e anonymous referees, and t h e publisher for helping us to produce this excellent collection of papers.

vi Ding-Zhu Du and Panos M. Pardalos University of Minnesota and University of Florida October 1992

Preface

vii

Contents Preface

v

Greedily Solvable Transportation Networks and Edge-Guided Vertex Elimination Ilan Adler and Ron Shamir

1

1. Introduction 2. Preliminaries 3. Vertex Elimination and Spanning Trees 4. Generating an O p t i m a l Spanning Tree 5. Gale Certificates 6. Signatures 7. S u m m a r y References Networks Minimizing Length Plus t h e N u m b e r of S t e i n e r P o i n t s Thomas Colthurst, Chris Cox, Joel Foisy, Hugh Kathryn Kollett, Holly Lowy, and Stephen Root

1 4 7 11 12 15 19 20

23 Howards

1. Introduction 23 2. Examples of Networks Minimizing Length Plus t h e N u m b e r of Steiner Points .24 3. Bounds on t h e N u m b e r of Edges Meeting at Steiner Points 32 References 35 P r a c t i c a l E x p e r i e n c e s U s i n g an I n t e r a c t i v e O p t i m i z a t i o n P r o c e d u r e for V e h i c l e S c h e d u l i n g Joachim R. Daduna, Miodrag Mojsilovic, and Peter Schiitze

37

1. Introduction 2. Problem Formulation 3. M a t h e m a t i c a l Formulation 4. Operational Process 5. Example 6. Results 7. Conclusions 8. Outlook References

37 38 40 43 43 48 49 51 51

viii

Contents

S u b s e t I n t e r c o n n e c t i o n D e s i g n s : G e n e r a l i z a t i o n s of S p a n n i n g T r e e s and Steiner Trees Ding-Zhu Du and Panos M. Pardalos

53

1. Introduction 2. Multi-Phase Spanning Networks 3. Multi-Phase Steiner Networks References

53 54 59 61

Polynomial and Strongly Polynomial Algorithms for C o n v e x N e t w o r k O p t i m i z a t i o n Dorit S. Hochbaum

63

1. Introduction 2. P r o x i m i t y Theorems and Piecewise Linear Approximations 3. T h e Impossibility of Strongly Polynomial Algorithms 4. Q u a d r a t i c Network Flow Problems References

64 68 77 80 88

H a m i l t o n i a n C i r c u i t s for 2 - R e g u l a r I n t e r c o n n e c t i o n N e t w o r k s Frank K. Hwang and Wen-Ching Winnie Li

93

1. Introduction 2. A General Approach 3. T h e E x t e n d e d Double Loop Network 4. Semi-Torus Networks 5. S e m i - M a n h a t t a n Networks References

94 95 96 103 106 109

E q u i v a l e n t F o r m u l a t i o n s for t h e S t e i n e r P r o b l e m in G r a p h s Bassam N. Khoury, Panos M. Pardalos, and Donald W. Hearn

Ill

1. Introduction 2. T h e S P G and t h e S P D G 3. Mixed Integer Formulations 4. Integer Formulations 5. Continuous Formulations 6. Concluding Remarks References

Ill 112 113 114 119 120 121

Contents

ix

M i n i m u m Concave-Cost Network Flow Problems with a Single Nonlinear Arc Cost Bettina Klinz and Hoang Tuy

125

1. Introduction 2. A P a r a m e t r i c Method for Rank Two Quasiconcave Minimization 3. A Special M i n i m u m Concave-Cost Network Flow Problem 4. A Strongly Polynomial T i m e Algorithm for SSU Networks 5. An Improved Algorithm for SSU Networks 6. S u m m a r y and Concluding Remarks References

125 127 130 137 138 143 143

A M e t h o d for S o l v i n g N e t w o r k F l o w P r o b l e m s with General Nonlinear Arc Costs Bruce W. Lamar

147

1. Introduction 2. Problem Formulation 3. Conversion Procedure 4. Numerical Examples 5. S u m m a r y References

147 150 152 158 165 166

A p p l i c a t i o n of G l o b a l L i n e S e a r c h in O p t i m i z a t i o n of N e t w o r k s Jonas Mockus

169

1. Global Line Search 2. Optimization of Networks 3. T h e Optimization of High-Voltage Net of Power System 4. Mixed Integer Global Line Search References

169 170 171 173 175

S o l v i n g N o n l i n e a r P r o g r a m s w i t h E m b e d d e d N e t w o r k S t r u c t u r e s . . . . 177 Mustafa C. Pinar and Stavros A. Zenios 1. Introduction and Background 2. T h e Linear-Quadratic Penalty Algorithm for Networks with Side Constraints and Variables 3. Numerical Experience 4. Conclusions References

177 179 189 200 200

x

Contents

O n A l g o r i t h m s for N o n l i n e a r D y n a m i c N e t w o r k s Warren B. Powell, Elif Berkkam, and Irvin J. Lustig

203

1. Introduction 2. T h e N D N - X Formulation 3. T h e Transformed Problem N D N - T 4. Solution Algorithms for N D N - T 5. Numerical Results 6. Appendix: Calculation of t h e Derivatives References

203 205 207 209 212 215 221

S t r a t e g i c a n d T a c t i c a l M o d e l s a n d A l g o r i t h m s for t h e C o a l I n d u s t r y U n d e r t h e 1990 Clean Air A c t Hanif D. Sherali and Quaid J. Saifee

233

1. Introduction 2. Related Models in the Literature 3. Formulation of a Long-Term Strategic Model 4. Solution Procedures 5. C o m p u t a t i o n a l Experience Appendix: Modifications for a Tactical Day-to-Day References

234 236 237 246 249 255 261

M u l t i - O b j e c t i v e R o u t i n g in S t o c h a s t i c E v a c u a t i o n N e t w o r k s J. MacGregor Smith

263

1. Problem Overview 2. Assumptions and Definitions 3. Mathematical Model 4. Congestion Properties 5. Algorithm 6. Example 7. S u m m a r y and Conclusions References

263 265 267 268 272 274 280 280

A S i m p l e x M e t h o d for N e t w o r k P r o g r a m s w i t h C o n v e x Separable P i e c e w i s e Linear Costs and Its Application to Stochastic Transshipment Problems Jie Sun, K.-H. Tsai, and L. Qi 1. Introduction

283

284

Contents

xi

2. Background Materials 3. T h e Simplex Algorithm for ( N e t P L P ) and Its Convergence 4. Implementation of t h e Algorithm 5. C o m p u t a t i o n a l Results 6. T h e S T P and Computational Results 7. S T P C and O t h e r Extensions References

285 288 288 292 294 296 298

A Bibliography on N e t w o r k Flow P r o b l e m s Marinus Veldhorst

301

1. Introduction References

301 304

Tabu Search: Applications and P r o s p e c t s Stefan Vofl

333

1. Introduction 2. Tabu Search 3. Applications 4. Concepts for Parallel Tabu Search 5. Conclusions References

333 334 338 346 350 351

T h e S h o r t e s t P a t h N e t w o r k a n d I t s A p p l i c a t i o n s in Bicriteria Shortest Path Problems Guo-Liang Xue and Shang-Zhi Sun

355

1. Introduction 2. T h e Shortest P a t h Network 3. Applications 4. Conclusions References

356 356 359 360 361

A N e t w o r k F o r m a l i s m for P u r e E x c h a n g e E c o n o m i c E q u i l i b r i a Lan Zhao and Anna Nagurney

363

1. Introduction 2. T h e Variational Inequality Model of t h e P u r e Exchange Economy and its Isomorphic Network Equilibrium Representation 3. A General Iterative Scheme for t h e C o m p u t a t i o n of Walrasian Price Equilibrium

363 365 367

xii 4. T h e Projection and Relaxation Methods for t h e C o m p u t a t i o n of t h e Equilibrium Prices 5. Numerical Examples 6. S u m m a r y and Conclusions References

Contents

372 380 384 385

S t e i n e r P r o b l e m in M u l t i s t a g e C o m p u t e r N e t w o r k s Sourav Bhattacharya and Bhaskar Dasgupta

387

1. Introduction 2. Multistage Interconnection Networks 3. Multistage Communication Networks 4. Conclusion References

387 389 397 400 401

1 Network Optimization Problems, pp. 1-22 Eds. D.-Z. Du and P.M. Pardalos ©1993 World Scientific Publishing Co.

Greedily Solvable Transportation Networks and Edge-Guided Vertex Elimination Ilan Adler IEOR Department,

University

of California,

Berkeley,

CA 94720

USA.

Ron Shamir Department of Computer Science, Sackler Faculty of Exact Sciences, University, Tel-Aviv 69978, ISRAEL.1

Tel Aviv

Abstract

The greedy algorithm for the transportation problem repeatedly picks an edge, maximizes flow on it and updates the supplies and demands. If, with the same order of edges, the greedy algorithm gives an optimal (resp., feasible) solution for every feasible supply and demand functions, we call that order an optimality (resp., feasibility) sequence. We show that with a feasibility sequence, one can guarantee a feasible solution which is also a spanning tree, or give a certificate on infeasibility. Furthermore, with an optimality sequence, one can guarantee an optimal basic solution for every feasible problem. We also show how to obtain a spanning tree with a given signature or a dual feasible basis with a given signature, using feasibility and optimality sequences. Our results build on some interesting properties of vertex elimination algorithms which are guided by edge orders.

1

Introduction

T h e transportation problem can be stated as follows: A commodity which is available at certain sources is demanded at some destinations. Shipping costs from each source 'Supported by AFOSR grants 89-0512 and 90-0008, and by NSF grant STC88-09648.

2

I. Adler & R.

Shamir

to each destination are known. Given the amount available at each source and the a m o u n t d e m a n d e d at each destination, the problem is to determine how much to send directly from each source t o each destination so t h a t all supplies and d e m a n d s are met and t h e total shipping cost incurred is m i n i m u m . W h e n not every source can ship to every destination, we say t h a t the problem is restricted. T h e transportation problem is one of t h e fundamental problems in combinatorial optimization, and has been studied intensively over t h e last fifty years. T h e excellent surveys [2] and [11] describe several algorithms for transportation and related network flow problems. One n a t u r a l approach for solving t h e transportation problem is by a greedy algorithm, guided by a predetermined order of t h e edges. This algorithm repeatedly picks t h e next edge in t h a t order, sends t h e m a x i m u m possible a m o u n t of flow along it and u p d a t e s t h e supplies and d e m a n d s accordingly. In general, this approach does not guarantee an optimal solution, but is often used to generate an initial feasible solution for more elaborate methods in unrestricted problems. T h e "north-west corner rule" and t h e " m i n i m u m CV, rule" are two well-known examples of this approach (see, e.g., [13].) T h e greedy algorithm has t h e obvious advantage t h a t it is very fast, hence t h e interest in identifying when it guarantees an optimal solution. Research has focused on cases when there exists a single order of the edges which - when used by t h e greedy algorithm - is guaranteed to produce an optimal solution for all feasible problems on t h a t network. Such order of the edges is called an optimality sequence. T h e knowledge of an optimality sequence is useful in particular if one needs to solve several problems with t h e same costs, but with varying supplies and demands: Given an optimality sequence, each problem is solvable in linear time. Similarly, if the greedy algorithm guided by some edge-order produces a feasible solution for every feasible problem, t h a t order is called a feasibility sequence. Hoffman [15] gave a characterization of bipartite networks which a d m i t an optimality sequence, in t e r m s of t h e Monge property. (The property will be defined in t h e next section.) Dietrich and Shamir [8, 18] (see also [19]) generalized the characterization to restricted transportation problems. Many families of problems which fit those characterizations or closely related ones have been studied in operations research, computational geometry and molecular biology applications. (See [1] for a list of references.) Alon et. al. [3] provided an efficient algorithm which finds an optimality sequence or determines t h a t no such sequence exists for unrestricted problems. Dietrich and Shamir [8, 18, 19]) generalized t h e algorithm to restricted transportation problems. Adler, Hoffman and Shamir [1] characterized t h e bipartite graphs which admit a feasibility sequence in graph-theoretic terms and gave very efficient algorithms for recognizing t h e m . They also extended t h e characterizations and algorithms for feasibility and optimality sequences to general, non-bipartite networks. In this paper we introduce a requirement t h a t the solution obtained from t h e greedy algorithm will also be basic. This requirement has several motivations: In m a n y linear p r o g r a m m i n g situations, t h e optimal solution of a problem is also required

Greedily

Solvable

Transportation

Networks

3

to be basic. This is needed when dual information is required, or for post-optimality analysis. Such complete information on t h e problem is not always available from an optimal solution, unless it is also a dual feasible basis. For example, a solution to t h e dual problem m a y not be immediately available in t h a t situation. In case t h e problem is infeasible, it may also be i m p o r t a n t to obtain a proof of infeasibility by pointing out a "bottleneck" which caused t h e infeasibility in t h e original problem. For example, m e t h o d s using linear programming solvers as subroutines (or "oracles"), require a separating hyperplane in case the problem is infeasible (cf. [10]). How much t h e n do we have to sacrifice in terms of complexity, and in t e r m s of t h e size of t h e class of greedily solvable problems, in order to guarantee t h a t we get a basis? As it turns out, by modest modifications of t h e algorithms, we can guarantee obtaining basic information without increasing t h e complexity and without decreasing t h e size of the class. It is well known in linear programming theory t h a t proper p e r t u r b a t i o n of t h e right-hand side a n d / o r the objective function generates a problem in which t h e only optimal solution is also an optimal basis. However, it is interesting to show how to achieve an optimal basis directly, by equipping an optimization algorithm with an appropriate mechanism (typically some sort of tie breaker). For example, Bland [7] showed how to prevent cycling in t h e simplex m e t h o d , by endowing t h e simplex algorithm with some combinatorial properties, instead of using t h e geometric 'trick' of p e r t u r b a t i o n . Similarly, for t h e greedy algorithm in t h e transportation problem, we construct here edge selection rules which guarantee independently t h a t t h e final selected set is a basis, primal feasible and dual feasible. One immediate consequence of this setup is t h a t imposing t h e intersection of these selection rules guarantees t h a t t h e final set of edges is an optimal basis. T h e "modular" approach outlined above yields additional results: Since these selection rules are independent, one can combine a subset of these rules with other rules so t h a t t h e final selected set of edges will satisfy additional desired properties. For example, an additional rule which enforces a given signature leads t h e greedy algorithm (using an optimality sequence) t o produce a dual feasible spanning tree with a a prescribed signature. This t y p e of analysis is used to obtain several additional results. Our m a i n results are the following: • In section 3 we define a very simple generic vertex elimination algorithm, which scans t h e edges in predetermined order and eliminates one vertex in each step. We show t h a t with any edge order and on any graph, t h a t algorithm generates a spanning forest. These results do not require the transportation and greedy algorithm setting, and may be of interest by themselves. Next we show t h a t for bipartite graphs, if the edge order forms a feasibility sequence, then t h e generic algorithm forms a spanning tree and therefore a basis. Moreover, if t h e order is an optimality sequence, then t h e resulting basis is dual feasible. • In section 4 we slightly modify t h e greedy algorithm so t h a t it will have t h e

4

I. Adler & R.

Shamir

properties of t h e vertex elimination algorithm defined in section 3. We show t h a t with an optimality sequence this new algorithm guarantees an optimal basis for every feasible problem. • In section 5 we concentrate on t h e consequences on negative termination in t h e greedy algorithm: Whenever t h e algorithm uses a feasibility sequence and determines t h a t a problem is infeasible, we can identify a "bottleneck" set of sources (or destinations) whose total supply exceeds t h e total d e m a n d of all their neighbors in t h e original problem. • As a by-product, we show in section 6 t h a t t h e tools developed here yield some interesting results on signatures of transportation problems. Recall t h a t t h e signature of a tree on a bipartite graph is t h e vector of t h e degrees of t h e sources in t h a t tree. We give an algorithm which when guided by a feasibility sequence, produces a spanning tree with a given signature or determines t h a t t h e signature is invalid. T h e same algorithm, when guided by an optimality sequence, produces a dual feasible basis with t h a t signature, if it is valid. Finally, we give new characterizations for feasibility and optimality sequences using signatures.

2

Preliminaries

Let G = (7, J; E) be a b i p a r t i t e graph whose two vertex sets (or sides) are I and J , where E C I x J , |7| = m , \J\ = n and \E\ = p. T h e vertices in I and J are numbered 1 , . . . , m and m + 1 , . . . , m -\- n, respectively. We assume throughout t h a t graphs are undirected and connected. For convenience, ij and ji are used interchangeably to denote t h e edge between i and j . Denote V = IUJ. T h e set N(v) = {j € V\vj 6 E} is called t h e neighborhood of v, and dv = \N(y)\ is called t h e degree of vertex v. v is called an end-vertex if dv = 1. For each edge ij 6 E, a cost Ci3 > 0 is assigned. Again, we use b o t h dj and Cji for t h e cost of t h e edge between i and j . Edge costs m a y also be represented by an m x n m a t r i x , in which case we define C;,- = oo for each ij $ E. T h e graph together with its edge-costs are called a network and are denoted N = (G,C). Finally, for each vertex v £ V, a non-negative excess e„ is given. T h e bipartite network together with an excess vector specify t h e input to a transportation problem. T h e problem is called restricted if t h e underlying bipartite graph is incomplete. T h e origin of t h e model is in planning of shipping, where t h e sets I and J correspond to sources and destinations, respectively. A commodity available at t h e sources must be shipped to satisfy d e m a n d s at t h e destinations, dj is t h e cost of shipping each unit of t h e c o m m o d i t y on edge ij. ev is t h e supply available at source v if v € I, or t h e d e m a n d at destination v if v 6 J- T h e transportation problem is to determine t h e a m o u n t Xij to be shipped each edge ij, so t h a t all supplied and d e m a n d s are met

Greedily

Solvable

Transportation

Networks

5

at m i n i m u m overall cost, i.e., min

s.t. JZ xa

= e,

i<EV

Xij

> 0

ij G E

(P(N,e))

We say t h a t t h e excess vector e is feasible if P(N, e) has a feasible solution, i.e., a solution in which all supplies and demands are satisfied. An obvious necessary condition for feasibility is t h a t J2iei e; = Y.j^j^jGiven a spanning tree T in G, t h e (unique) solution of YlijeT xij = e > ? £ ^ xij = 0 ij £ E — T is called t h e the primal basic solution associated with T. (A spanning tree is sometimes called a basis, since t h e set of columns corresponding to its edges forms a m a x i m a l independent set in t h e coefficient m a t r i x of t h e equations in P(N,e).) If t h e primal basic solution is non-negative t h e n it is called a feasible solution, and T is called primal feasible. T h e dual of t h e transportation problem is defined as : 2 J eiui + zZ is/ jeJ S.t. U; + Vj < Cij

max

e v

jj (D(N,e)) ij G E

Given a spanning tree T in G, t h e (unique) solution of t h e system ui + Vj = Cij ij G T is called t h e the dual basic solution associated with T. If t h e dual basic solution satisfies t h e rest of t h e inequalities in D(N,e) then it is called dual feasible solution, and T is called dual feasible. A tree which is b o t h primal and dual feasible is called optimal. Let S = (Si,S2,.-.Sp) be a p e r m u t a t i o n of t h e edges. For a problem P(N,e), t h e greedy algorithm maximizes each variable in t u r n , according to the order given in S. A formal description is the following. (We assume t h a t initially X;J = 0 for all ij G E). algorithm G R E E D Y ( S ) ; begin For i = 1, ...,p do : begin pick the next edge •?,- = rs. xrs «- min{e r ,e 5 } 6r
- ij -< jk is impossible (see figure 1). We say that S has the Hoffman property for the network ./V if the following property holds (see figure 1): The Hoffman Property: For every i,k 6 I and j , I € J, such that ij, il, kj and kl are all edges in s .. s E, if ij -< il and ij -< kj then Cij + Ckt < Cu + CkjA sequence is called a Monge sequence for the network N if it satisfies both the Z property and the Hoffman property. Note that in a complete bipartite graph every permutation trivially satisfies the Z property, hence the Hoffman property and the Monge property are equivalent on unrestricted transportation networks. The following theorem summarizes some of the known characterizations and complexity results for optimality and feasibility sequences in bipartite networks. (Recall that a bipartite graph is called chordal bipartite if it does not contain an induced cycle of length six or more [12].) For the complexity results we assume m < n: Theorem 2.1 (a) [15, 19] S is an optimality sequence for a network if and only if it is a Monge sequence.

Greedily

Solvable

Transportation

Networks

1

(b) [19, 1] S is a feasibility sequence for G if and only if it has the Z property. (c) [1] A bipartite graph admits a feasibility sequence if and only if it is chordal bipartite. (d) [3, 19] In a bipartite network, one can construct an optimality sequence or determine that no such sequence exists in 0(pm log n) steps. (e) [1] In a bipartite graph, one can construct a feasibility sequence or determine that no such sequence exists in 0(p\ogn) steps.

3

Vertex Elimination and Spanning Trees

First, we consider vertex elimination in a general setting. A basic step which will be used in several algorithms in t h e sequel is eliminating a vertex together with all t h e edges incident on it from t h e graph G(V, E). A formal description is t h e following: procedure ELIMINATE^); begin V - rt -< pt, so by t h e Z property of t h e sequence S, pj £ Ek~x. But then also pj £ Ek, so j £ Lk. Hence all neighbors of r in Gk~x are also present in Gk, which implies Dk~x = Dk U {rt}. This immediately implies (ii). (i) follows since e * " 1 ^ - 1 ) = ek{Rk)

+ e*" 1 > ek(Lk)

+ e*" 1 = ek~\Lk^),

(5.1)

where t h e inequality follows from t h e inductive hypothesis. C a s e II: r £ I, t £ Lk. Here we want to show t h a t Rk = Rk~1, i.e., rt is not in t h e connected component of w in Bk~l. This will imply (i) and (ii). Suppose there exists a p a t h in Bk~x from r to w. Such p a t h must contain an edge it £ Bk_1 such t h a t i is connected to w in Bk~x, since r is eliminated in step k — 1. B u t then it £ Bk and i £ Rk, so t £ Lk, a contradiction. C a s e III: r £ J, t g Rk. In t h a t case Rk = Rk~1, and r £ Lk, since r g Vk. (i) and (ii) are thus i m m e d i a t e by t h e inductive hypothesis. C a s e I V : r £ J, t £ Rk. By t h e inductive hypothesis, t belongs to t h e same component as w in Bk. Hence in I ? * - 1 , both r and t belong to t h a t component, which implies t h a t there exists an edge ti on t h e p a t h from t to w in Bk~1, and t h a t edge is also present in Bk. Since every other neighbor of t (except r ) is also in Vk, by t h e inductive hypothesis it is also in Lk. Hence (ii) is true, (i) now follows by an argument identical to t h e chain of inequalities (5.1) in case I. • Applying t h e proposition with k = 0 we get the desired result: T h e o r e m 5.2 If S is a feasibility sequence and algorithm GREEDY* detects infeasibility at vertex w, then the vertices in w 's connected component of B which are on the same side as w form a Gale certificate of infeasibility. •

Greedily

6

Solvable

Transportation

Networks

15

Signatures

Let B b e a subset of t h e edges in t h e bipartite graph G = (7, J; E). In this section it will be more convenient to denote edges by ordered pairs of vertices, i.e., for t h e edge (r, s) G E, r € I and s G J. T h e (row) signature of B is defined as an integer vector crB = ( o f , . . . , o-%) such t h a t

a? = \{k\(i,k)eB}\

« = l,...,m.

(6.1)

T h e column signature is defined analogously. We shall discuss row signatures here, b u t t h e same results apply to column signatures, by interchanging t h e roles of I and J. For a bipartite graph an integer vector )L « 2 + 2.732L. 2. One Steiner point connecting three of t h e vertices and a side of t h e square connecting t h e fourth. This has cost C(L) = 1 + (1 + ^ + &)L « 1 + 2.932L. 3. One Steiner point at t h e center of t h e square connecting all four vertices. This has cost C(L) = 1 + 2^/2L « 1 + 2.8281. 4. T h e perimeter m e t h o d , with no Steiner points and three sides of t h e square. Here t h e cost C(L) = 3L. T h e network with two Steiner points is t h e unique network with two Steiner points and four vertices [C2]. We see here t h e scale dependence: when L > /?^ / i , , t h e length minimizing network is cost-minimizing. For * ,- < L < ^ r 1 /-_ , t h e third network with four edges meeting at t h e center is minimizing. T h e perimeter m e t h o d is best for relatively small values of L, when L < * y-. L e m m a 2.2 In a network minimizing length plus the number of Steiner points, sum of the unit vectors of segments meeting at a Steiner point must be 0.

the

PROOF: For a given vector v, t h e derivative of |v| is A . T h e sum of t h e derivatives, i.e., of t h e unit vectors meeting at a point m u s t add t o zero, or else we could move t h e Steiner point to a position t h a t does allow t h e unit vectors to sum to 0, and t h e network would have reduced total length. • In particular, any Steiner point of degree three must have 120° angles between its edges. L e m m a 2.3 Suppose a network N(0) is continuously transformed to a new network N(l) of the same topological type. Then the function F(t) describing the length of the intermediate networks is strictly convex. PROOF: Consider any two networks ./V(O) and ./V(l) of a given topological t y p e , with nodes rti(0), ...,nm(Q) and n ^ l ) , . . . , n m ( l ) . T h e n a n e w network N(t) of t h e same t y p e can be created by interpolating between each pair of nodes n,(0) and ra,-(l) to form the node n , ( i ) . Let F(t) equal the length of N(t), which will clearly be a function of t. T h e n

m = £ fuM ni,rij

26

T. Colthurst

et a}.

where fij(t) — d(ni(t),rij(t)), t h e distance between two connected nodes «,,«,•. Consider two pairs of corresponding points ra,(0),rjj(0) and n ; ( l ) , n j ( l ) . T h e n .1. / l j (

2)

n,-(0) + n,-(l) n,-(0) + n,-(l) =

d(

2

'

2

>

+

nj(l)\\

so fiA\)

= \hi(0)+ni{l),nj(0)

Considering t h e four points as vectors, it follows from t h e triangle inequality t h a t i | K ( 0 ) + n,(\),n3(0)

+ ^(1)11 < | ( | | n , - ( 0 ) + ^(0)11 + ||r»,-(l) + ^ ( l ) | | ) / . , ( 0 ) + /.•,(!) 2

Hence fi,j(\) < ,; 2 which implies t h a t /( 2, the cost-minimizing network connecting the 2ra vertices ( 0 , . . . , ± a , ...,0) of a regular cross-polytope with at most one Steiner point connects every vertex directly to the origin for a > J—T-—^—T=T. PROOF: Consider t h e cross-polytope of edge length / = a\/2. Since it has In vertices, connecting t h e m by a tree on t h e perimeter has cost l(2n — 1). Connecting its vertices to a Steiner point at t h e origin has cost 1 + y/2nl. T h u s , t h e center network is cheaper for / > l/(2ra — \/2n — 1) and for such an I has cost greater t h a n 1 + _ ^* _, • Any cost-minimizing network N with one Steiner point may be classified by t h e n u m b e r of vertices t h e Steiner point connects; call this n u m b e r t. Since t h e Steiner tree portion of N has length at least 4 - t h a t of the m i n i m u m spanning tree [GH], N has cost at least 1 + l[2n — t) + -yz(t — 1)/. Comparing this to the center cost of 1 + y/2nl

one concludes t h a t it is cheaper to connect a vertex to the Steiner point if

n(2-\/2)—7-

t < —-^T—iS-.

For t higher t h a n this, we claim t h a t there is a vertex not directly

connected to t h e Steiner point which is closer to it t h a n to t h e nearest vertex. We m a y assume t h a t t h e In — t unconnected vertices are m u t u a l l y adjacent, containing no two vertices on the same axis, and form a 2n — t — 1 dimensional simplex. For if neither of two points on the same axis are connected to t h e Steiner point, symm e t r y dictates t h a t t h e Steiner point m u s t lie in the hyperplane orthogonal to their axis. Since it m u s t b e contained in t h e convex hull of the vertices in t h e hyperplane,

Networks Minimizing Length Plus the Number of Steiner Points

27

the Steiner point will actually be closer to both vertices than they are to any other vertices. On the other hand axis directions with both vertices connected directly to the Steiner point need not be considered either, as the Steiner point will always lie in the K" _1 dimensional orthogonal hyperplane, so the problem can be considered in Call the distance between the origin and the Steiner point x. Then the coordinates of the Steiner point look like (-£-, ~fri ~TTi •••)"7§-)i a " d the coordinates of a given unconnected vertex are ( /,""_,, 0,0, ...0). Then the distance between them is just x

I

(2n-Qa»

fVlVl'

2n-t

Hence, a vertex is closer to the Steiner point than to other vertices if f >

2 XI ^ + 2x2 V2n-i

x < /(2,/2 - —?

7=L=)

Clearly x is less than the distance from a vertex to the origin, so we know that x
1.5 which is true for all n > 2. Thus, the network connecting all the vertices at the center is the cheaper than networks of all other topological types with one Steiner point. It remains to show that the Steiner point should be placed at the origin in the minimal network of this type. Consider how the length of the central network changes as the Steiner point is moved away from the center along some vector u. The change in length with respect to time is given by ^ = — (uj + v2... + v6) • u where the V{ are the six unit vectors along the axes. A quick calculation show that 4j = 0 for all u, so the origin is an equilibrium for this configuration. By Lemma 2.3 length is a strictly convex function of the position of the Steiner point, so it follows that the network connecting all the vertices to the center must be minimal. •

28

T. Colthurst

L e m m a 2.5 The unit vectors in the six axis directions minimizing network for the octahedron.

in 5K3 constitute

et a.1. a cost-

PROOF:

By L e m m a 2.4 we know t h a t t h e unit vectors are cost-minimizing over all networks with (at most) one Steiner point. We will show t h a t there exists a scaling (a range of /) over which t h e unit vectors are of lesser cost t h a n t h e minimal spanning tree and all possible trees with two or more Steiner points. T h e minimal spanning tree has cost 5y/2l and t h e unit vectors have cost 6/ + 1. It can be easily calculated t h a t if / > 0.934 > >s_6> t h e n t h e unit vectors will be minimizing over t h e minimal spanning tree. Now consider networks having at least two Steiner points. T h e cost of such a network is at least two plus t h e length of t h e shortest network connecting all six points. A group of Williams students in 1989 proved t h a t t h e length-minimizing network for the octahedron had four Steiner points. T h e y also showed t h e eight combinatorial choices for t h e minimizer [CI], by classifying t h e topological types according to how t h e four Steiner points connect pairs of vertices on t h e same axis. Here we examine each case individually. All we have to do is find a lower bound for each candidate, then show t h a t for some / > 0.934, t h e network with one Steiner point is cheaper. Note t h a t in these representations, x, y, and z represent t h e axis t h a t each segment hits at distance 1 from t h e origin. Case 1:

z

X

Figure 2: Case 1. T h e network at right has length three times t h a t of t h e piece of t h e network component on t h e left, which has length > 1.93/. This case uses a Steiner point at the origin which is connected to t h e vertices in pairs using three additional Steiner points. Hence every pair of vertices sharing an

Networks

Minimizing

Length

Plus the Number

of Steiner

Points

29

axis are connected through three Steiner points. Here a straightforward c o m p u t a t i o n shows t h a t t h e length is greater t h a n 5.7/, which means t h a t when 6/ + 1 < 5.7/ + 2, t h e one Steiner point network will be cheaper. T h u s for 0.934 < I < 3.333, t h e one Steiner point network will be better. Cases 2, 3, 4:

Figure 3: Case 2. T h e portion of t h e network in t h e dotted circle has length at least 21.

Y

Z

Figure 4: Case 3. T h e ?/-legs have length > 2/ and t h e remaining portion of t h e network is t h e minimizing network on a square with side length \/2l.

Figure 5: Case 4. T h e z-legs have length > 2/ and t h e remaining portion of t h e network is t h e minimizing network on a square with side length \J2l. Cases 2, 3, and 4 consider t h e possible topological types in which exactly one pair of boundary points on the same axis are connected through a single Steiner point. T h e m i n i m u m distance between this pair of points is 21. T h e cost of these three cases is bounded below by 2/ + L, where L is the length of t h e minimizing network connecting four points on t h e same plane forming a square with side length y2l. More specifically, t h e overall length is bounded by ( 4 w | + \/2 — \ + 2)1. T h u s , it is

T. Colthurst et al.

30

cheaper to use one Steiner point when 6/ + 1 < ( 4 w | + v 2 —yr + 2)1 + 2, i.e. when 0-934 < / < 6_(4(0,g11)+1.7g) » 1.04. Cases 5, 6:

Figure 6: Case 5. The length of the portion of the network in each dotted circle is > 21.

'•--/.is-'

X-i..--'

Figure 7: Case 6. The length of the portion of the network in each dotted circle is >2l. Cases 5 and 6 are the two topological types in which all three pairs of vertices on the same axis are connected through a single Steiner point. In these cases, it is clear that cost is bounded by 6/. Clearly, the one Steiner point network is cheaper. Case 7:

Figure 8: Case 7. The parts of the network in the dotted ellipses each have length > 21, and the parts of the network in the dotted circles each have length > i / | . In this case two pairs are connected through two Steiner points and the other through four. Its cost is bounded by (2 -f 2 + 2 J\l,

a n d t h e parts in t h e large ellipse have length > ( 4 - i / | + \ / 2 — -\)l-

In this case two pairs of vertices are connected through two Steiner points a n d t h e other through four. Its length is bounded by t h e shortest network connecting x, x, y a n d y + 2 i / | / . In other words, by 6 ^ / | + y/2 — -7g. T h u s , t h e 1 point network is b e t t e r when 0.934 < 7 < 6_{6{J1)+0,i8) < 1.78. D We have seen t h a t four edges of a network t h a t minimizes length plus t h e n u m b e r of Steiner points may meet in 5R2, and t h a t six edges may meet in 5R3. We will now complete t h e proof t h a t for n > 2, there is a network in 3?" t h a t minimizes length plus t h e n u m b e r of Steiner points and has In edges meeting at a point. T h e o r e m 2 . 6 The In vectors of length a in the In axis directions constitute a costminimizing network for some value of a. In fact, if n > 2, then the unit vectors are cost-minimizing. PROOF: By L e m m a 2.4, we need only consider networks with two or more Steiner points. A network with two or more Steiner points must again have length of at least -j- t h a t of t h e m i n i m u m spanning tree [GH], and thus cost of at least 2 + -T;(2?I — 1)/, where / = \/2a is t h e distance between adjacent points. B u t for I < l / ( v 2 n — 4 - ( 2 n — 1)) this is greater t h a n t h e cost for t h e center network, a n d for n > 5, l/(\/2n —7T(2TC — 1)) > l / ( 2 n — \/2n — 1), the value for / at which t h e center network becomes cheaper t h a n t h e minimal spanning tree. Therefore there exists a range of values for a over which t h e vectors along t h e axes form a minimal network for n > 5. For n = 2 a n d n = 3 , Examples 2.1 a n d L e m m a 2.5 provide examples in which t h e vectors along t h e axes form cost-minimizing networks. For n = 4, t h e cross-polytope has t e t r a h e d r a as 3-faces, and any network on it m u s t have length twice t h a t of t h e shortest network on a tetrahedron, or 2(2.445)/ [Cl], since we may project it onto a tetrahedron in four orthogonal directions. T h u s any network with two or more Steiner points has cost > 2(1 + 2.445), which is more expensive t h a n t h e center network for l / ( 2 n - V^n - 1) = 0.744 < / < 1.319.

32

T. Colthurst

et al.

T h u s for n > 4 there is a range of / for which 2n edges of length / meeting at a point is cost-minimizing. •

3

Bounds on the Number of Edges Meeting at Steiner Points

T h e o r e m 3.5 establishes four as t h e sharp upper bound on t h e n u m b e r of edges in a cost-minimizing network t h a t can meet at a Steiner point in the plane. Proposition 3.6 and Corollary 3.7 establish lower bounds on t h e n u m b e r of edges t h a t can meet in 5K". First L e m m a 3.1 gives a lower bound on t h e angle between two edges in a cost minimizing network. L e m m a 3.1 If AB and AC are two edges meeting at a point A in a network, then IB AC > 60°.

cost-minimizing

P R O O F : If LB AC < 60°, then LABC or IBCA > 60°. W i t h o u t loss of generality, take LABC > 60°. T h e n ||AC|| > \\BC\\, so AB + BC is a cheaper network t h a n AB + AC which was, by assumption, minimal. • R e m a r k 3 . 2 An exponential Steiner point.

upper bound on the number of edges that can meet at a

L e m m a 3.1 yields as an upper bound on the n u m b e r of segments t h a t m a y meet at a point t h e n u m b e r of balls of radius | t h a t can be packed on t h e unit sphere. This upper bound is 6 in 3J2, between 12 and 14 in 3J3, and grows exponentially as a function of n.

Figure 10: In a triangle with largest angle 7, ||a|| + ||6|| > |(cj|(2cos-7 + 1).

Networks

Minimizing

Length

Plus the Number

of Steiner Points

33

a4

Figure 11: Five edges cannot meet at a Steiner point in a cost minimizing network. L e m m a 3 . 3 Given a Aabc such that 90° > 7 > a and 7 > /? (see Figure 10),

then

||a|| + | | 6 | | > | | c | | ( 2 c o s 7 + l ) . PROOF: W i t h o u t loss of generality, fix 7 and take /3 > a. By t h e law of sines,

Nl sin a

=

II&II sin j3

Ml

=

=

sin 7

HI sin(/? + 7)

Therefore ||q|| + ||6|| ^ s i n ( / ? + 7 ) + s i n ^ ||c|| sin7 T h e m i n i m u m value of this expression when varying f3 occurs when /? = a , so

W^i

||c||

= l+ ^

sin 7

= 2cos7+l.

D L e m m a 3.4 If five edges, po,Pi, • • • ,Pi, meet at a Steiner point in a cost-minimizing network in the plane, as in Figure 11, then a; < 2 t a n - 1 ;A=, for all i € { 0 , 1 , . . . , 4 } .

PROOF: W i t h o u t loss of generality, let Lao — 29 be t h e largest angle. By L e m m a 2.2 t h e unit vectors must add to zero. Consider t h e components in t h e po and p$

34

T. ColthuTst

et al.

directions: 2 c o s # + cos(# + ai)-|-cos(# + a:i + oc2) + cos(a4 + 8) = 0. 6 is maximized if a a = Q 2 = an = 60° by aconvexity argument, so 2cos 6> + cos(# + 60°) + cos(6l + 120°) = 0, or 6 = t a n - 1 ^ , so a 0 = 20 < 87.795°. a T h e o r e m 3.5 Four is a sharp bound on the number point in a cost-minimizing network.

of edges meeting

at a

Steiner

PROOF: We have shown examples of four edges meeting (See Examples 2.1). By L e m m a 3.1 it follows t h a t seven or more edges cannot meet, and if six meet all angles must be exactly 60°. In this case, however, a regular hexagon can be substituted in some small region about the Steiner point, and if one edge is removed the length is decreased. This creates six new Steiner points, but they can be eliminated by expanding t h e hexagon until t h e new Steiner points reach t h e vertices at t h e end of t h e original six edges. For this reason, six edges cannot meet. Hence it remains to determine if five can meet. Consider five edges p 0 , . . . , p 4 meeting at a Steiner point in a cost-minimizing network. ct0 > /30 and a0 > f0, since otherwise we could switch a0 for p0 or p 4 and decrease t h e overall length. Now apply L e m m a 3.4 to obtain ||p,|| + ||pi+i]| > | | a ; | | ( 2 c o s a , + l ) . T h e n 2 £ 4 = 0 | | p ; | | > £ 4 = 0 ||a,||(2cos a, + 1). W i t h o u t loss of generality, let a0 be t h e longest side. T h e cost of t h e perimeter is Z ] 4 = 1 ||'II + !• Since X^=o 0, so 1 1 4 -|| 0, and i£||a1||(2cosa, + l)>£||a,||. Z

i=0

."=1

4

Recall t h a t 2 £ = o ||p,|| > Et=o ||a.||(2cos a,- + 1), so

EW>EWt=0

i=l

and

Ellwll + i>ElNli=0

i=l

Networks

Minimizing

Length

Plus the Number

of Steiner

Points

35

But t h e n t h e cost of five edges meeting at a Steiner point is greater t h a n t h e cost of t h e perimeter. Therefore, five edges cannot meet at a Steiner point in a cost minimizing network. D P r o p o s i t i o n 3 . 6 If t points are distributed on a unit n-sphere such that the minimum angle a between any two points is greater than 78.687947° and ift > 1/(1— i//ft/»)' where fi = A/2 — 2 cos a is the minimum distance between two points, then for some scaling to a sphere of radius r, there is a cost-minimizing network in which at least a third of the points meet at a Steiner point. PROOF: If t h e n-sphere has radius r then connecting the points through t h e center has cost 1 + r t . Since t h e m i n i m u m distance between any two points is y/2 — 2 cos ctr = fir, connecting t h e points through a perimeter tree has cost at least fir(t — 1). T h u s t h e center network has lower cost for r > l/(fi{t — 1) — t), which is positive for t > 4 and a > 78.68°. All other one Steiner point networks can be determined by t h e n u m b e r of points they connect; call this number c. Since t h e Steiner tree portion of t h e network must have cost at least A- times the minimal spanning tree [GH], the one Steiner point network has m i n i m u m cost 1 + l/%/3/3r(c — 1) + fir(t — c), which is greater t h a n 1 + rt for c
y =l i=i

X>y = l J'=I

xo-€{0;l} r nach

Ankunf t

MWZ F-Art

Einsparung Zeitpuffer

MAX

BBF

6:43

0

N

1

_< 6:59 SPE 7:24>_ HID

BBF SRE

7:11 7:36

0 0

V N

-3 8

BBF

8:13

0

N

7:46

SRE

Vorschlaege seit der letzten Einsparung:

0

Veraend.

1

Koennen Sie eine Veraenderung urn 3 Minuten akzept .eren ?

Figure 5.1: Suggestion for alteration shown on the scree In t h e first step t h e following suggestion for alteration appears on screen (see Fig. 5.1): T h e trips shown, in this case four, represent a segment from one block. Lines 5 and 6 contain t h e d a t a of those two trips which may be linked after altering the respective d e p a r t u r e times. Lines 4 and 7 show in addition t h e trips before t h e first and after t h e second trip to be linked. Such an additional information helps t h e user to assess more easily t h e impacts of an alteration. T h e following will be indicated: t h e number of t h e line on which t h e trip will be carried out, d e p a r t u r e and arrival time, first and last stop of t h e trip, (in-company) delay buffer ( M W Z ) , the type of trip ( F - A r t ) , and t i m e range (Zeitpuffer) for alterations or the missing time for a link. This would mean in t h e example above t h a t trip 1 starts at 6:59 a.m. on line 131 from stop S P E and ends at B B F at 7:11 a.m. V (the type of trip) indicates t h a t it is an additional trip. (The other types of trips, mentioned in t h e following, are: N = regular trip, Y = school trip.) T h e value -3 in t h e field "Zeitpuffer" of trip 1 at 6:59 indicates t h a t a link with trip 2 is not possible, because there is a shortage of three minutes between t h e end of t h e first trip and t h e start of the second one. True, we have thirteen minutes between t h e two trips, b u t there is not enough t i m e for a deadhead trip from B B F to HID, plus layovers. However, subsequent to trip 2, there is still a t i m e range of eight minutes in S R E (after subtracting two minutes for layover). True, we have t h e possibility to start trip 2 t h e necessary three minutes later. But trip 2 is a trip with an even headway, published in t h e timetable, and therefore in general not p e r m i t t e d to be shifted.

46

J.R. Daduna,

M. Mojsilovic,

and P.

Schntze

— Fahrzeuge: Anfangswert 350

Linie

Abfahrt

gegenwaertige Anzahl 349

von —> nach

Ankunft

MWZ F-Art

Einsparung Zeitpuffer

1

Veraend.

davor Fahrt 1 Fahrt 2

275 275

danach

275

_< 6:43 SKA 7:08>_ RAD 7:32

MEO

RAD MEO

7:07 7:29

0 0

N N

RAD

7:54

0

N

Vorschlaege seit der letzten Einsparung:

1

Koennen Sie eine Veraenderung um

-1 0

1

1 Minuten akzept .eren ?

Figure 5.2: Second suggestion for alteration F u r t h e r m o r e , there is a connection to City Rail in B B F which has to be adhered to. Trip 1, on t h e contrary, is an additional trip and therefore not published in t h e timetable. From an in-company point of view, this makes it possible to start this t r i p t h e required three m i n u t e s earlier. Should t h e suggestion be accepted, then already one vehicle has been saved in this example. Often, however, further alterations will have to be m a d e until a saving is a t t a i n e d , because a restructuring of t h e basis solution usually requires several steps. T h e suggestion for alteration shown in Fig. 5.2 will be indicated in t h e next iteration. In this case, t h e line above trip one contains no data. T h i s m e a n s t h a t trip 1 represents t h e start of t h e block. Correspondingly, trip 2 forms t h e end of a block, if no information after this trip is given. This suggestion for alteration cannot be accepted for in-company reasons, because it is a regular t r i p with a fixed headway. Only one alteration to t h e d e p a r t u r e t i m e is suggested in Fig. 5.1. This was t h e only possibility in t h e earlier versions of t h e sensitivity analysis. By expanding t h e p r o g r a m m e system, we can now shift the two trips which have to be linked. This is described by taking t h e above example one step further. After not having accepted the suggestion in Fig. 5.2 nor three other suggestions for alteration, t h e user will be presented with t h e suggestion shown in Fig. 5.3. For linking the school trips at 7:02 a.m. and 7:17 a.m., there is a shortage of four m i n u t e s . Due to t h e d a t a structure, t h e departure time of one of t h e two trips might be altered by t h e required four minutes. However, it is appropriate in this situation to shift both trips by two minutes.

Practical

Experiences

for Vehicle Scheduling

Fahrzeuge: Anfangswert 350

Linie

Abfahrt

47

g e g e n w a e r t i g e Anzah] 349

von —> nach

Ankunf t

Einsparung Zeitpuffer

MWZ F-Art

1

Veraend.

davor Fahrt 1 Fahrt 2

293

danach

700

293

_< 7:02 WSS 7:17>_ WSS 7:45

USU

SZU SZU

7:12 7:30

0 0

Y Y

ULS

7:53

0

Y

Vorschlaege s e i t der l e t z t e n Einsparung:

Koennen S i e e i n e Veraenderung um

-4 4

5

4 Mixiuten akzept .eren ?

Figure 5.3: Fifth suggestion for alteration in t h e second iteration.

At first, t h e user must enter whether he accepts the suggestion. be asked to indicate a shift for one of t h e two trips. This is shown t h e n u m b e r "2" before t h e sign "_©t--ooo\

Figure 7.1: Distribution curve of t h e n u m b e r of vehicles ( V H H ) T h e examples of HHA and V H H , two companies which have used t h e sensitivity analysis for several years, show t h a t cost-saving potentials result even from, to some extent, small alterations to the timetable in successive schedule periods. T h e cause can be found in the complexity of t h e problem structure. Even small alterations may allow entirely new combinations. T h e m a i n objective for the sensitivity analysis is, also because of its incorporation in t h e planning process, t h e reduction of the m a x i m u m fleet size. This, however, does not m e a n t h a t t h e usage has to be limited to the t i m e before and after t h e absolute peak, which in the case of VHH (see Fig. 7.1) lies between 5:30 a.m. and 8:40 a.m. Because t h e number of vehicles required at certain times determines t h e n u m b e r of duties to some extent, it might be of interest with regard to d u t y scheduling to analyse also t h e local m a x i m a . W i t h a distribution curve of t h e n u m b e r of vehicles like t h e one of C T B (see Fig. 7.2), which in total has four almost equal m a x i m a , all peaks m u s t be analysed with regard to t h e m a x i m u m number of vehicles. By altering an originally absolute m a x i m u m , it is possible with such a structure t h a t due to a

Practical

Experiences

for Vehicle

Scheduling

51

i50j Nujnbe]- 0 f busses

oo

©

o

C4

o

oo

O

oo

IllMMNMIIIIII iiillllllll © © O O © O «n ^£5 r - 00 2 for all i. (\Xi\ with \Xi\ = 1 can always be deleted.) (c) Every feasible graph G satisfies G = Uj^jG,-. In fact, an edge not in U J ^ G ; can be deleted without changing t h e feasibility. To see an example, consider five subsets X\ = { f i , ^ } , ^ = { " l i ^ j ^ 3 ) 1 -^3 = {i>3,^4,v{\, X4 — {vi,V2,v4}, and Xs = {v2,v4,Vs}. These subsets and t h e weight function w are as shown in F i g . l . Some feasible graphs for (Xi,X2,X3,X4,Xs) are in Fig.2. Among t h e m , G* is a m i n i m u m feasible graph for (w ; X\,Xv, A3,X4,Xs).

w 1 2

1

2

3

4

5

5

6

7

g

5

6

7

5

6

3 4

5

5

Figure 1: An input of t h e M P S p N

Figure 2: Feasible graphs for t h e input of Fig.l T h e M P S p N is an N P - h a r d problem. It is closely related t o several classical problems. For m = 1, t h e M P S p N is exactly t h e m i n i m u m spanning tree problem. G r a h a m [13] gave an interesting historical survey. T h e most popular polynomial-time algorithm for t h e m i n i m u m spanning tree problem was discovered by Kruskal [15]. A set system (X\, • • • ,Xm) t h a t has a feasible graph to be a forest is called subtree hypergraph. Such a system has various applications in computer science [1] and statistics [16]. It is also related to chordal graphs [9][10]. How can we tell whether a set system is a subtree hypergraph? Tarjan and Yannakakis [21] found an 0(m -f rc)-time algorithm.

56

D.-Z. Du and P. M. Pa.rda.los

If (X\,- • • ,Xm) is a subtree hypergraph, then we can find a m i n i m u m feasible graph for (w ; X j , - • • ,Xm) in polynomial time. In fact, a subtree hypergraph (Xi, • • • ,Xm) must satisfy t h e following condition. (A) T h e r e exists a feasible graph G such t h a t for any i, j 6 {1, • • •, m } , G[X;] fl G[X,] is a tree, where G[X,] is t h e subgraph of G induced by X{. T h e proof is easy. Suppose t h a t G is the feasible forest. Consider two vertices u and v in G[X,-] 0 G[X,]. Since G is feasible, there exists a p a t h in Gi from u t o w and there also exists a p a t h in G[Xj] from u to v. However, G is a forest. So, t h e two paths are identical, which is a p a t h in G[X;] fl G [ X , ] . T h u s , G[X,] fl G[X,] is connected. A connected subgraph of a forest must be a tree. Therefore, (A) holds. T h e following theorem has been proved in [6] for t h e unit-weight case. We will show t h a t it is also t r u e in general. T h e o r e m 2.1 If (X\, • • •,Xm) satisfies condition (A), then computing feasible graph for (w ; X\, • • • ,Xm) can be done in polynomial time.

a

minimum

Before doing so, let us first look at an efficient special case given by Du and Miller [6]. Consider t h e following condition. (B) For any two subsets X,- and Xj, if |X; n Xj\ > 2, then there exists a subset Xk such t h a t Xi 0 Xj = XkThis condition implies t h a t for any 7 C {1, • • •, TO} either | C\iejX{\ < 1 or there exists k £ {1, • • • ,m} such t h a t Xk = flig/X,-. T h e following theorem was shown in [6]. T h e o r e m 2.2 If (B) holds, then all complements independent subsets of a matroid.

of feasible graphs form a family

of

^,From this theorem and t h e theory of matroids [12], we see t h a t t h e M P S p N with (B) can be solved by t h e following two greedy algorithms. A L G O R I T H M 1. G := t h e complete graph of X; Sort edges of G in weight-increasing order, e\, • • •, e„( n _i)/ 2 ; for i := 1 to n(n — l ) / 2 do if G \ e; is feasible then G : = G \ e;. A L G O R I T H M 2.

Subset

Interconnection

57

Designs

Sort all edges of t h e complete graph of X in weight-decreasing order, ei,- • • ,e„(n-i)/2; G:=0; for i := 1 to n(n — l ) / 2 do if there exists j such t h a t e, connects two connected components of G[-Xj] then G := G U e,-. Algorithm 2 is a variation of a greedy algorithm given by Prinsner [19]. It was proved in [6] t h a t if (A) holds, t h e n t h e feasible graph in (A) is a m i n i m u m feasible graph for (1; X\, • • •, Xm) and every m i n i m u m feasible graph for (1; X\, • • •, Xm) possesses t h e property in (A). In addition, all m i n i m u m feasible graphs for ( 1 ; X\,- • • ,Xm) are m i n i m u m feasible graphs for (1; CiieiXi, I C {1, • • •, m } ) . It follows t h a t : T h e o r e m 2.3 If (A) holds, then all minimum form a family of bases of a matroid.

feasible

graphs for

(l;Xi,---,Xm)

This theorem means t h a t t h e M P S p N with (A) can also be solved by greedy algorithms like Algorithms 1 and 2. T h u s , Theorem 2.1 is proved. An application of T h e o r e m 2.1 is to study t h e M P S p N with small m. It was proved by Tang [11] t h a t for m = 1,2,3, (A) holds. So, we have: C o r o l l a r y 2.4 The MPSpN

is polynomial-time

computable for m = 1,2,3.

Since the M P S p N is equivalent to a problem about intersection of m matroids, t h e case for m = 3 is a little surprising. In fact, a problem about three matroid intersection is usually N P - h a r d [11]. Tang [11] also constructed a counterexample which shows t h a t (A) does not hold for m > 4. T h e following question, therefore is still open. O p e n P r o b l e m 1 With a fixed m > 4, is the MPSpN

polynomial-time

computable?

Note t h a t in t h e unit-weight case, it is easy to show t h a t for m fixed, M P S p N is polynomial-time solvable. So, t h e problem is significant only for general weight function. An extension of bounding m is t h e following condition. (a,/?) for any a distinct points x-i, x?, • • •, xa £ X, there are at most f) X^s containing all X\, X2-, ' ' ' , Xa where a and (j are two natural numbers. Clearly, ( a , /?) implies (a1, /3') whenever a' > a and /?' > /3. Under t h e assumption t h a t all weights equal one, Du [8] proved t h a t for a < 2 and /? < 2, t h e problem is polynomial-time solvable while for a > 3 or j3 > 6 or (a > 2, /? > 3) t h e problem is N P - h a r d . T h e following is an open problem

left in [8].

58

D.-Z. Du and P. M.

Pardalos

O p e n P r o b l e m 2 What is the computational complexity of computing a feasible graph for (1; X\, • • •, Xm) in the case of a = 1 and 3 < (3 < 5 ?

minimum

We believe t h a t it is polynomial-time solvable in case ( a = 1,(3 = 3), b u t is N P - h a r d in cases ( a = 1,(3 = 4) and ( a = l,/3 = 5). T h e interesting work in [8] is about t h e case of a = (3 = 2. T h e problem is transfered to a m a x i m u m matching problem on a graph. It is not known whether this work could be extended to arbitrary edge-weights. An application of the results on a = (3 = 2 is to construct a p p r o x i m a t e solutions for t h e M P S p N . There are several ways. T h e first is to divide t h e collection of subsets [Xi ,••• , Xm) into several small collections satisfying condition (a = 2, (3 = 2), construct a m i n i m u m feasible graph for each small collection, and then union all of t h e m . T h e second is as follows: W h e n a pair of points appears in more t h a n two subsets X^s, we stick t h e m together. In this way, we can reduce t h e original collection of subsets to a new one, satisfying condition (a = 2, (3 = 2). After a m i n i m u m feasible graph for t h e new collection of subsets is found, we break stuck pairs by adding some edges. Although several heuristics have appeared in t h e literature, none of t h e m has been proved to be a bounded heuristics; i.e., a heuristic t h a t produces an approximation solution with total weight within a constant factor from optimal. T h u s , t h e following is an i m p o r t a n t question for the M P S p N . O p e n P r o b l e m 3 Does the MPSpN

have a bounded

heuristic?

Du and Miller [6] proved the following. T h e o r e m 2 . 5 Any set system (Xi, • • •,Xm) can be partitioned in time 0(mn) less than \Jlm subsystems such that for each (X,-,, • • •, X^) of them, X^, • • •, Xii an element in common.

into have

W h e n all weights equal one, a m i n i m u m feasible graph for (1; X\, • • •, Xm) with n^LjX,- ^ 0 is t h e star with vertex set X and t h e center chosen from flJ^A",-. This fact with Proposition 2. together yields a \/2m-heuristic for t h e M P S p N in t h e unitweight case. However, t h e same idea does not work for t h e M P S p N in general. To see this, we point out t h e following. T h e o r e m 2.6 Computing r\?=1Xi jt 0 is NP-hard.

a minimum

feasible

graph for (w ; X\,-

• •, Xm)

with

To prove this, let us mention an N P - h a r d problem, the vertex covering: Given a graph G, find a smallest subset of vertices t h a t cover all edges [11]. To do polynomialt i m e reduction, choose a point s different from all vertices and for each edge uv, give a set Xuv = {v,u,s}. Moreover, we assign weights in t h e following way: t h e distance between any two vertices of G is one; t h e distance from s to any vertex of G is n2

Subset

Interconnection

Designs

59

where n is t h e n u m b e r of vertices of G. Now, we claim t h a t there is a vertex covering of size at most k if and only if there is a feasible graph of total weight at most

\\G\\ + kn2. In fact, if a vertex covering of size at most k exists, then connecting s to every vertex in t h e vertex covering, we obtain a feasible graph of weight at most ||G|| + kn2. For t h e other direction, suppose t h a t a feasible graph of weight at most ||G|| + kn2 exists. In such a graph t h e degree of s must be at most k. Otherwise, t h e total weight would exceed ||G|| + kn2. Now, all vertices adjacent to s in t h e feasible graph form a vertex covering of size at most k. T h u s , we proved t h a t t h e vertex covering can be polynomial-time reduced to t h e considered problem in T h e o r e m 2. If t h e endpoints of an edge belong t o more X^s, then they are more likely to appear in a m i n i m u m feasible graph. From this thought, Prisner [19] gave t h e concept of "benefit". For a graph G and a set system (X\, • • •, Xm), t h e benefit b(uv,G) of an edge uv is t h e n u m b e r of G[X;]'s such t h a t uv connects two connected components of G[X;]. T h e benefit-cost ratio of uv is b(uv, G)/w(uv) where w(uv) is weight of t h e edge uv. Using this concept, Prisner [19] discovered a YA=2 7-heuristic for t h e M P S p N where K is the m a x i m u m number of X^s, which have two elements in common. His heuristic runs in t i m e 0(n4 + mn2) as follows. A L G O R I T H M 3. G:=0; while t h e r e is an edge of positive benefit do choose an edge uv with largest benefit-cost ratio and set G := G U uv. Prisner's heuristic is the best known heuristic for t h e M P S p N .

3

Multi-Phase Steiner Networks

Given an edge-weighed graph B with vertex set X and subsets X\,Y\, • • • ,Xm, Ym of X with Xi r\Y{ = 0, t h e problem considered in this section is to find a m i n i m u m weighed subgraph G such t h a t for every i = l , - - - , m , G contains a Steiner tree for Xi without using vertices not in Y{. We will call this problem t h e Multi-Phase Steiner Network Problem ( M P S t N ) and its solution a minimum feasible graph for [w ; X\, Y\, • • •, Xm, Ym]. For m = 1, the M P S t N is t h e m i n i m u m Steiner tree problem [22] which is already NP-hard. T h u s , t h e M P S t N is much harder t h a n t h e MPStN. It is a well-known fact [14] t h a t t h e m i n i m u m spanning tree is a 2-approximation of t h e m i n i m u m Steiner tree; i.e., the total length of a m i n i m u m spanning tree is not bigger t h a n twice t h e total length of a m i n i m u m Steiner tree. However, there does not exist a constant c such t h a t t h e total weight of a m i n i m u m feasible graph for (w • X\, • • • ,Xm) is not bigger t h a n c times t h e total weight of a m i n i m u m feasible graph for [w ; X\, Y\, • • •, Xm, Ym\. To see this, we look at t h e following example.

60

D.-Z. Du and P. M. Pa.rda.los

Let Xij = {i,j} and Yij — {i,j,0} for 1 < i < j < n. A m i n i m u m feasible graph for (1; Xij, 1 < i < j < n) t h e n has weight n(n — l ) / 2 . However, a m i n i m u m feasible graph for [1; Xij,Yij, 1 < i < j < n] has weight n. T h u s , t h e ratio between t h e m is (n — l ) / 2 , which cannot be bounded by a constant. W h a t is t h e relation between m i n i m u m feasible graphs for (w ; Ylt • • •, Ym) and [w ; X\, Yx, •••, Xm, Ym]l In t h e unit-weight case, Chao and Du [3] proved t h e following. T h e o r e m 3.1 If G is a minimum feasible graph for (1 ; Y\, • • •, Ym) satisfying condition (A), and for every edge uv of G there exists an index i such that u,v 6 Xi, then G is a minimum feasible graph for [1 ; X-y, Yj, • • •, Xm ,Ym]. So far, t h e following question is still open. O p e n P r o b l e m 4 Could Theorem

3.1 hold for general weight

function?

Concerning the ratio between m i n i m u m feasible graphs for (w ; Y\,---,Ym) and [w ; Xi, Yi, • • •, Xm, Ym], we look at t h e following example. Let X\ = {1,2} and Y\ = {1, 2, • • •, n}. T h e n a m i n i m u m feasible graph for (1; Yi) must have weight n — 1 while a m i n i m u m feasible graph for [1; X\, Y\\ has weight 1. Such a ratio cannot have a constant bound. In t h e above example, t h e m i n i m u m feasible graph for (1; Yi) contains many unnecessary edges for [1; A ^ Y i ] . Such edges are easily deleted. So, t h e following question arises: O p e n P r o b l e m 5 Starting from a minimum feasible graph for (w ; Y\,- • • ,Ym), could we modify it in polynomial time to obtain a bounded approximation solution of minimum feasible graph for [w ; Xi, Y\, • • • ,Xm,Ym}? In [6], two operatuions for simplifying a feasible graph for (1; Xi, • • • ,Xm) were described. Such operations can be extended to simplify a feasible graph G for [w; X1,Y1,---,Xm,Ym]as follows. ( 0 1 ) Check each edge e of G. If t h e removal of edge e preserves t h e feasibility of t h e graph G, then delete e. ( 0 2 ) Check each edge e of G, the complement of G. If we can use operation ( 0 1 ) to remove some edges of e U G with total weight more t h a n the weight of e, then remove such edges and add e. Clearly, t h e two operations work in polynomial time. An open question left in [6] is as follows. O p e n P r o b l e m 6 Can we obtain a bounded approximation of the minimum feasible graph for (1; Xi, • • • , Xm) from the minimum feasible graph for (1; fl.-g/X,-, / C {!,••• ,m}) by using operations (01) and (02)?

Subset Interconnection Designs

61

We can ask a similar question between [w ; X\, Y\, • • •, Xm, Ym] and [w ; Pl.-g/i^, / C {1, •••,"»}]. An interesting special case for the MPStN is n^ljY; ^ 0. In this case, there exists a star which is a feasible graph. Many problems in the real world can be decomposed into several subproblems in such a special case. So, a feasible graph can be obtained by taking the union of solutions of subproblems. Unfortunately, it was proved in [7] that: T h e o r e m 3.2 Computing a minimum feasible graph for [1; Xi,Y\, • • • ,Xm,Ym] n^jK' + 0 is still NP-hard.

with

For [1; Xi, Y\, • • •, Xm,Ym] with n^LjY; j= 0, the star feasible graph has, at most, twice the weight of a minimum feasible graph. However, it is not known whether or not for [w ; X\,Y\, • • • ,Xm,Ym] with n^jYi 7^ 0 there is a bounded approximation with a factor two from optimal. Bern and Plassmann [2] proved that the minimum Steiner tree problem in graphs is MAX SNP-hard; i.e., it is unlikely to have a polynomial-time approximation scheme. The MPStN clearly inherits this property.

References [1] C. Beeri, R. Pagin, D. Maier, M. Yannakakis, On the desidability of acyclic database schemes, J. ACM30 (1983) 479-513. [2] M. Bern and P. Plassmann, The Steiner problem with edge lengths 1 and 2, Information Processing Letters 32 (1989) 171-176. [3] S.-C. Chao and D.-Z. Du, A sufficient optimality condition for the valveplacement problem, J. North-East Heavy Industry Institute, 4 (1983) (in Chinese). [4] D.-Z. Du and Y.-M. Chen, Placement of valves in vacuum systems, J. Electric Light Sources, 4 (1976) (in Chinese). [5] D.-Z. Du, An optimization problem, Discrete Appl. Math., 14 (1986) 101-104. [6] D.-Z. Du and Z. Miller, Matroids and subset interconnection design, SIAM J. Disc. Math., 1 (1988) 416-424. [7] D.-Z. Du and X.-F. Du, A special case of valve-placement problem, Acta Mathematics Applicatae Sinica, 4 (1991) (in Chinese). [8] D.-Z. Du, On complexity of subset interconnection designs, DIMACS TR 91-23.

62

D.-Z. Du and P. M.

Pardalos

[9] P. Duchet, Propriete de Helly et problemes de representation, in: Colloque ternational Paris-Orsay 260 (1978) 117-118. [10] C. F l a m e n t , Hypergraphes arbores, Discrete

Mathematics

21 (1978) 223-227.

[11] M.R. Garey and D.S. Johnson, Computers and Intractability, Theory of NP-Completeness, Freeman, San Francisco, 1979. [12] M. Gondran and M. Minoux, Graphs and Algorithms, 1984.

In-

A Guide to the

John Wiley, New York,

[13] R.L. G r a h a m , P. Hell, On t h e history of t h e m i n i m u m spanning tree problem, Ann. History Computing 7 (1985) 43-57. [14] R.M. K a r p , Reducibility among combinatorial problem, in R.E. Miller and J . W . T h a t c h e r (ed.), Complexity of Computer Computation (Plenum Press, New York, 1972) 85-103. [15] J . B . Kruskal, On the shortest spanning subtree of a graph and t h e traveling saleman problem, Proc. Amer. Math. Soc. 71 (1956) 48-57. [16] S.L. Lauritzen, T.P. Speed, K. Vijayan, Decomposable graphs and hypergraphs, J. Austral. Math. Soc. A 36 (1984) 12-29. [17] E. Prisner, Intersection-representation of graphs in n-cyclomatic graphs, Combinatoria, to appear.

Ars

[18] E. Prisner, Familien zusammenhangender Teilgraphen eines Graphen und ihre Durchschnittsgraphen, Dissertation in Universitat H a m b u r g 1988. [19] E. Prisner, Two algorithms for the subset interconnection design, Networks, appear.

to

[20] R.E. Tarjan, M. Yannakakis, Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Comput. 13 (1984) 566-579. [21] T.-Z. Tang, An optimality condition for m i n i m u m feasible graphs, Applied ematics, 2 (1989) 21-24 (in Chinese). [22] S. Vofi, Steiner

Probleme

in Graphen, (Hain, Frankfurt, 1990).

Math-

63 Network Optimization Problems, pp. 63-92 Eds. D.-Z. Du and P.M. Pardalos ©1993 World Scientific Publishing Co.

Polynomial and Strongly Polynomial Algorithms for Convex Network Optimization Dorit S. H o c h b a u m 1 School of Business Administration Research Department, University

and Industrial Engineering of California, Berkeley

and

Operations

Abstract

Numerous recent developments in complexity theory and the design and analysis of algorithms have contributed to new insights into nolinear network flows problem. In particular, it is now known that the separable convex costs network flow problem is polynomial, in either integers or continuous variables, the latter for output with specified number of significant digits. Moreover, assuming a reasonable computation model consisting of the arithmetic operations and the floor operation and comparisons, it was demonstrated that strongly polynomial algorithms are impossible for the problem. It follows that the available polynomial algorithms are very close in their running time to optimal algorithms. Several properties of convex separable network problems are investigated here. These include so-called proximity theorems that establish the proximity of integer and continuous solutions, and also that of the closeness of an optimal solution to a scaled piecewise linear approximation of the nonlinear problem to the optimial solution to the nonlinear problem. Such proximity theorems make it possible to use scaled piecewise linear approximations in algorithms of polynomial complexity and obtain integer solutions from continuous and vice versa. Classes of algorithms that use the proximity properties are described. Quadratic network problems have some special properties differentiating them from other nonlinear instances of the problem. It is shown how these properties could be exploited for faster and more efficient algorithms. Issues related to quadratic nonseparable network problems are discussed as well. Complexity issues and NP-hardness of classes of nonlinear network flow problems are presented and important directions for future work are delineated.

64

1

D. S. Hochbaum

Introduction

The general nonlinear costs network flow problem is an instance of nonlinear optimization subject to linear constraints. Given a network G = (N,A), let T be the n x m node-arc adjacency matrix of the network, n = \N\ and m = \A\, x the vector of flows on the arcs, x = {a;.j}(ij)e4, &v the supply (demand) of node i, and U;J the capacity upper bounds. The integer formulation of the nonlinear flow problem is, min F(x) s.t. Tx = b 0 < x < u x integer For F(x) a linear function, the problem of minimum cost flows have been studied since the early 60's, and has been studied extensively in the past decade. Until recently, results for the nonlinear problem were limited to iteratively convergent algorithms, rather than algorithms with finite established complexity. An extensive review of such iterative algorithms for the continuous nonlinear constrained problem is given in [Min86a]. The reason for the lack of finite algorithms is that in general the continuous version of the problem (when the integrality requirement is omitted) may posses solutions that cannot be expressed as finite output. Such solutions are irrational numbers or even nonalgrbraic numbers that cannot be represented as a solution to a certain polynomial equation. This property renders complexity theory inapplicable, as it requires finite length input and output. The nonconvex continuous version of the problem is NP-hard [Sah74] even for the quadratic case and so are many simplified versions of the nonseparable problem even if convex. The corresponding concave minimization problem is in general NP-hard. The complexity of the concave network problem as well as some algorithms (exponential) are discussed in [GP90], [GP91a] and [GP91b]. An excellent unifying presentation of some polynomial instances of the concave flow problem is given in [EMV87]. There the problem is proved polynomial when the arcs are uncapacitated and number of demand nodes is fixed. It is also proved polynomial for certain classes of planar graphs. Yet, even with the assumption of convexity, a quadratic nonseparable problem when the set of constraints on the adjacency matrix is empty, is NP-complete. (An example is provided in section 4). It is therefore the case, that for general purpose polynomial algorithms we focus primarily on the class of problems with F{x) = £ fij{xij) where fij are convex. Throughout we discuss convex network flow problems with the classification: 1. Separable or nonseparable 2. Quadratic or nonquadratic and nonlinear 3. Integer or continuous. Hochbaum and Shanthikumar proved several proximity results for convex separable nonlinear optimization ([HS90]). For the nonlinear separable network flow

Polynomial

&c Strongly

Polynomial

Algorithms

for Convex Optimization

65

problem these d e m o n s t r a t e closeness properties between an integer optimal solution to t h e problem and a continuous optimal solution, between two scaled solutions to t h e problem, and between a scaled solution and an o p t i m a l (integer or continuous) solution. A scaled solution is a solution to t h e problem when t h e objective is given on a piecewise linear grid determined by t h e scaling constant. T h e closeness in all these case is t h a t t h e two vectors differ in t h e m a x i m u m n o r m by at most m or 2m units. E d m o n d s and K a r p ' s [EK72] approach of capacity scaling could also be viewed as a proximity result, except t h a t for t h e m i n i m u m cost network flow problem, t h e solution at each phase is not necessarily a feasible solution to t h e scaled problem. We shall discuss this approach in t h e context of proximity. Defining what it means to obtain a solution in t h e continuous case is not straightforward due to t h e potential irrationality of t h e o u t p u t for nonlinear problems. We therefore use t h e definition of t- accurate solution, where log ^ is t h e n u m b e r of significant digits desired in t h e o u t p u t . A solution, x^ is e-a.ccura.te if there exists an optimal solution x* such t h a t ||a;' £ ' — a;*||oo < e. T h a t is, t is t h e accuracy required in t h e solution space. W h e n t h e functions f<j are convex nonlinear functions, both t h e continuous and integer versions of t h e problem are solvable by algorithms running in a n u m b e r of steps t h a t is a polynomial function of m and n, b u t also of log 2 ||M||OO or log 2 ||6||ooSuch an algorithm for t h e integer case is described by Minoux [Min86] using t h e capacity scaling approach, and for the integer and continuous cases in [HS90], using t h e proximity results, where t h e running t i m e for t h e continuous case depends polynomially on t h e n u m b e r of digits required to describe t h e solution. These algorithms are polynomial b u t not strongly polynomial, as they depend on t h e magnitudes of t h e n u m b e r s appearing in t h e problem instance, as well as t h e problem size p a r a m e t e r s . On t h e practice side, a heuristic implementation of this algorithm, using the interior point m e t h o d , indicates t h a t it is very efficient, [HS91]. T h e topic of strong polynomiality became an i m p o r t a n t issue once a polynomial algorithm, t h e Ellipsoid m e t h o d , was devised for solving linear p r o g r a m m i n g problems. T h e ellipsoid m e t h o d , as well as all other polynomial algorithms known for linear programming, is polynomial b u t not strongly polynomial. T h a t is, t h e running t i m e depends on t h e d a t a coefficients, rather then only on t h e n u m b e r of variables and constraints. Consequently, solving linear programming problems with different degrees of accuracy in t h e cost coefficients results in different running times. T h a t is, t h e actual n u m b e r of arithmetic operations grows as t h e accuracy of t h e d a t a , and hence the length of t h e numbers in the input, increases. Such behavior of an algorithm is undesirable as it requires careful monitoring of t h e size of t h e n u m b e r s appearing in t h e d a t a describing t h e problem instance, and hence limits t h e efficient applicablity of t h e algorithm. W h e t h e r linear programming is solvable in strongly polynomial t i m e is still an open problem, yet much progress has been reported. T h e most i m p o r t a n t work in

66

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Hochbaum

this context is t h a t of Tardos, [Tard86], t h a t established t h a t "combinatorial" linear p r o g r a m m i n g problems, those with a constraint m a t r i x having small coefficients, are solvable in strongly polynomial time. In particular, m i n i m u m cost network flow problems, which have coefficients t h a t are either 0, 1 or -1 in t h e constraint m a t r i x , are solvable in strongly polynomial time, [Tard85]. For non-quadratic objective functions, it was proved in [Hoc90] t h a t there is no algorithm solving t h e single source separable convex Transportation problem in strongly polynomial t i m e . This lower bound result holds either in the comparison model or in t h e algebraic computation tree model with t h e four arithmetic operations + , — , - , : , and comparisons, and it holds even if t h e floor operation is added. Since the single source Transportation problem is a special case of separable convex minimization problems over totally unimodular constraint matrices or over polymatroidal constraints, then in particular all nonlinear convex network flow problems cannot be solved in strongly polynomial time. T h e question of whether there exists a strongly polynomial algorithm for quadratic separable network flow problems remains open. T h e lower bound result in [Hoc90] does not apply to this case. Moreover, for quadratic problems there is a polynomial bound on the length of the o u t p u t (this is because the optimality conditions are linear). Partial results have been derived recently. Tamir, [Tam89], devised a strongly polynomial algorithm for m i n i m u m convex quadratic separable cost flow when the underlying network is series-parallel, in t h e presence of a single source-sink pair. In [CH90] a strongly polynomial algorithm for t h e Q u a d r a t i c Transportation problem with fixed n u m b e r of sources or sinks is described. T h e strongly polynomial algorithm described in t h a t paper delivers optimal continuous solutions using t h e operations mentioned above, without t h e floor operation. For t h e integer version, a strongly polynomial algorithm follows from a technique proposed in [HS90] t h a t relies on t h e proximity result. T h e technique is a strongly polynomial algorithm for deriving an integer optimal solution from a continuous optimal solution to nonlinear convex problems over totally unimodular constraint matrix. This technique requires the floor operation. In [HH92], Hochbaum and Hong describe a strongly polynomial algorithm for t h e q u a d r a t i c separable convex network Allocation problem with complexity equal to t h a t of a single m a x i m u m flow algorithm. This problem is not a general network flow problem as only some of t h e edges have costs associated with t h e m . All other edges only have capacity limits. Again, t h e idea here is to solve the continuous problem optimally, and then use the technique for converting a continuous optimal solution to a integer optimal solution. T h e r e is no hope of solving t h e integer version in strongly polynomial t i m e without t h e use of t h e floor operation. This follows from an observation by Tamir, [Tam89], illustrating t h a t the floor operation is essential for solving integer quadratic problems. Tamir has demonstrated this via t h e following quadratic Allocation problem which

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m a y be viewed as a quadratic Transportation problem with a single source node: min S.t.

[-x\

+ - ( a - l)a^]

Xi + X2 = b xi,

x2 > 0, integer

T h e optimal value of x2 is [-J. Therefore t h e floor operation can be executed via a routine t h a t solves a quadratic Transportation problem. This, along with t h e impossibility result of strongly polynomial algorithms for non-quadratic problems mentioned above, implies t h a t the floor operation, if helpful in devising strongly polynomial algorithms, is limited in its power to quadratic problems only. Although separable quadratic problems m a y be solvable in strongly polynomial t i m e , t h e question of strong polynomiality of nonsepamble quadratic continuous optimization problems subject only to nonnegativity constraints, is at least as hard as t h e question of strong polynomiality of linear p r o g r a m m i n g (this insight is due to I. Adler). Therefore, we do not investigate this issue here, as it should be treated in t h e framework of t h e strong polynomiality of linear programming. Nonseparable quadratic integer problems are N P - h a r d , [MK87], so in t h a t sense also these problems are beyond t h e scope of our current investigation. Nonseparable convex continuous problems, as well as nonseparable q u a d r a t i c convex problems are solvable as follows. A solution approximating t h e o p t i m a l objective value to t h e convex continuous problem is in principle obtainable in polynomial t i m e , provided t h a t t h e gradient of t h e objective functions are available and t h a t t h e value of t h e optimal solution is bounded in a certain interval. Such work, based on t h e Ellipsoid m e t h o d , is described by Yudin and Nemirovsky ([NY83]). In t h e q u a d r a t i c case, exact solutions are possible. Indeed, t h e polynomial solvability of continuous convex quadratic programming problems over linear constraints was established as a byproduct of the ellipsoid algorithm for Linear P r o g r a m m i n g (see Kozlov, Tarasov and Khachian, [KTK79]). T h e best running time reported to d a t e is by Monteiro and Adler [MA89], 0(m3L), where L represents t h e total length of t h e input coefficients and m t h e n u m b e r of variables. Similar results were also given by Kapoor and Vaidya [KV86]. T h e case for t h e integer problems t h a t are nonseparable is harder. We give in Section 5 a condition for such problems to be solvable in polynomial t i m e for t h e convex nonseparable problem. We t h e n describe special cases of integer nonseparable network flow problems t h a t are polynomial, all of which with quadratic objective. [HSS92] gave a polynomial algorithm for a specific quadratic nonseparable Transportation problem in integers. T h a t algorithm is strongly polynomial with some restrictions on t h e size of t h e linear coefficients in t h e objective function. Granot and Skorin-Kapov [GSK90] consider also a case of Transportation problem in integers solvable in polynomial t i m e . Cases where quadratic convex nonseparable optimization over box constraints (which are a special case of t h e quadratic network flow with an e m p t y set of flow

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balance constraints), have been shown to have strongly polynomial algorithms are: by Barahona [Bar86], by Baldick and Wu [BW90] and by Hochbaum [Hoc89]. All these are special cases in which the quadratic matrix in the objective can be made separable by using a totally unimodular transformation of the variables. When this is the case there is a polynomial time algorithm for separable convex optimization over totally unimodular constraints, that delivers optimal integer solutions in polynomial time ([HS90]). Baldick ([Bal91]) has also presented several classes of matrices that can be diagonalized (i.e. the quadratic problems are made separable) via the use of totally unimodular matrices. The plan of this paper is as follows. In section 1 we describe the scaled piecewise linear approximation and the proximity theorems for the separable flow problem and a general purpose polynomial algorithm. In Section 2 we propose a framework for obtaining several classes of algorithms that rely on these properties and on scaling. Section 3 provides the description of the lower bound proof that results in the impossibility of strongly polynomial algorithm for convex network flow problems. In Section 4 we demonstrate several results concerning the case of quadratic separable network flow, and Section 5 gives some limited results for nonseparable problems. The notation in this paper, includes bold letters for denoting vectors, and the notation e is used for the vector (1,1, ..., 1).

2 2.1

Proximity Theorems and Piecewise Linear Approximations The Scaled Piecewise Linear Approximation

Let fij : R —> R, (i,j) 6 A, be m convex functions and define

(•J')6A

We are interested in the solutions to the nonlinear convex integer network flow, (INF), problem: (INF)

min s.t.

F(x) Tx = b 0< x < u x integer

and to its continuous (real) relaxation {RNF)

min F(x) s.t. Tx = b 0< x < u

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69

Here T is an n x m adjacency m a t r i x of t h e network G = (V, A), 6 is a demandsupply n -vector, and u t h e capacity upper bounds vector on t h e arcs. T h e idea of linearizing a nonlinear function in order to obtain solutions is not new. In a 1959 book [Den59], Dennis writes regarding quadratic cost networks: "The electrical model for network flow problems can be extended to include flow branches for which the total cost contains terms depending on the square of the individual branch flows.... It appears that the algorithms presented in this chapter could be generalized.... These methods however are combinatorial in character and could require prohibitive calculation time, even on relatively simple networks. Certainly the simplicity and elegance of the diode-source algorithms would be absent. It would seem that the most practical means of attacking flow problems with quadratic costs would be to approximate the cost curve with piece-wise linear curve and substitute an appropriate number of linear cost branches connected in parallel." Whereas it is clear t h a t solving t h e problem on a piecewise linear approximation yields a feasible solution, t h e quality of such solution and its closeness to an optimal solution was not evaluated. [HS90] analyzed and described these proximity properties between the piecewise linearized problem's optimal solution and an optimal solution for t h e convex separable minimzation over linear constraints. We focus here only on t h e special case of network flow problems. We now describe formally t h e piecewise linear approximation. For this we introduce t h e following class of problems. For any scaling constant s € R+ let t h e scaled problem (RNF — s) be defined by

(RNF-s)

min

F(sy) Ty = b/s 0 < y < u/s

By setting x = sy in (RNF — s) it is easily seen t h a t (RNF — s) is t h e same as (RNF). If we define the integer scaled problem to be (RNF — s) with an integerality requirement t h a t y is integer, then there is a feasible solution only if b/s is integer. In [HS90], t h e constraints are given as inequality constraints. T h e scaled problem is then,

(INF-s)

min

F(sy) Ty > b/s -Ty > -b/s 0 < y < u/s y integer

Note t h a t a solution to (INF — s) is not necessarily feasible for t h e original problem. T h e a m o u n t of unsatisfied supply (demand) is bounded by n s units. This problem can be solved as a linear network flow problem by rounding t h e scaled supplies

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down to t h e nearest integer, and t h e scaled d e m a n d s ( t h a t are negative numbers) up to t h e nearest integer and adding a d u m m y node t h a t absorbes t h e difference between t h e total supply and total d e m a n d at cost 0. W h e n s = 1, the difference betweent the scaled supply and d e m a n d is 0, and hence there is no need for using t h e d u m m y node. E d m o n d s and Karp [EK72] used such idea of capacity scaling for t h e m a x i m u m flow problem t h a t can be formulated as a m i n i m u m cost problem with 6 = 0. Since t h e network flow problem readily provides integer solutions for integer right hand sides, and 0 is integer, feasibility was not an issue. Instead of t h e scaled version of (RNF) we now use a linearized version of (RNF) as defined below. For any s > 0 let /,^ : s : R —> R be t h e linearized version of fij such t h a t ffj's takes t h e same value as fij at all integer multiples of s: t h a t is, fij''s(sy) = fij(sy), for V integer and

S

~T~

x

S

S

iy

S

o

where [^-J (the floor of ^LL) is t h e largest integer value smaller t h a n or equal to ^-. Clearly fffs is a piecewise linear function which is convex if fij is convex. /,^ :s is depicted in Figure 1. Now define

{LNF-s)

min

FL'(x) Tx>b 0< x < u

where

Note t h a t t h e optimal solution to t h e integer program

(INF'-s)

min

FL:s{sy) Ty > b/s -Ty > -b/s 0 < y < u/s y integer

is also an optimal solution to (INF — s) because FL:s and F take t h e same value at integer multiples of 5 : i.e., FL:s(sy) = F(sy) for all y integers. Hence, to solve (INF — s) one can solve instead t h e piecewise linear version, (INF' — s). T h e set of feasible solutions {a; | Tx = 6,0 < a; < u] is bounded in a box of length B — minfJIuHoo, ||6||i} in each dimension (the flow on each edge cannot exceed capacity, or total sum of demands). In t h e initial iteration we substitute t h e tightest bound for t h e upper bound on t h e capacities. In t h e proximity-scaling algorithm, the

Polynomial

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Figure 1: upper and lower bounds on the variables will be updated at each iteration. So in a generic iteration t h e capacity bounds are replaced by constraints: Ijjj

2i -Eij _^ U ij i

(i,j)

G A.

These constraints will be scaled as well. In our procedure, at each iteration we shall work with t h e upper and lower bounds, U{j and Ljj, and a scaling constant s, such t h a t t h e length '^-, is an integer constant. We denote N= Each variable ?/;J, for t h e integer case, is then substituted by a sum of TV 0-1 variables:

., N. For t h e continuous case the substitution for i,-

c, = *{F i j+i>i; ) } s

k=i

o < 2,!*> < l So now (LNF — s) and (INF' on t h e variables z,( 2. In particular, t h e complexity of identifying an e-accurate single real root in an interval [0,.R] is O(loglog—) even if t h e polynomial is monotone in t h a t interval. Let Pi(x), ..., pn(x) be n polynomials each with a single root to t h e equation pi(x) = c in t h e interval [0, —], and each Pi(x) a monotone decreasing function in this interval. Since t h e choice of these polynomials is arbitrary, t h e lower bound on finding t h e n roots of these n polynomials is fi(rcloglog ~). Let fj(xj) = JQ3 pj(x)dx. T h e fjS are then polynomials of degree > 3. T h e problem, (P £ )

max

^2fj{xj-e) i

+

£J

B

j=i

t

c-xn+l-e

Xj > 0 Xj integer ,

has an optimal solution x such t h a t y — e • x is t h e (ne)-accurate vector of roots solving t h e system ' Pi(2/i) = c P2(«/ 2 ) = C

I Pn{Vn) = C. This follows directly from t h e Kuhn-Tucker conditions of optimality, and t h e fact t h a t an optimal integer solution to t h e scaled problem with a scaling constraint, s, x", and t h e optimal solution t o t h e continuous problem y* satisfy ||as* — J/*||oo < ns (Theorem 2.3). Hence, a lower bound for t h e complexity of solving (Pe) is Q(n log log -7-). For t = 1, we get t h e desired lower bound for t h e integer problem. In [MST88] there is a lower bound proof for finding epsi/on-accurate square roots t h a t allows also t h e floor, | J , operation. In our notation t h e resulting lower bound for our problem is fi(\/loglog ®), hence even with this additional operation t h e problem

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cannot be solved in strongly polynomial t i m e . Again, t h e quadratic objective is an exception and t h e algorithms for solving t h e quadratic objective simple resource Allocation problems rely on solving for t h e continuous solution first, then rounding down, using t h e floor operation, and proceeding to c o m p u t e t h e resulting integer vector to feasibility and optimality using fewer t h a n n greedy steps. See for instance, [IK88] for such an algorithm. Since t h e lower bound result applies also in t h e presence of t h e floor operation, it follows t h a t t h e "ease" of solving t h e quadratic case is indeed due to t h e quadratic objective and not to this, perhaps powerful, operation.

4

Quadratic Network Flow Problems

As noted earlier, t h e quadratic problem takes a special place among nonlinear optimization problems over linear constraints. This is because t h e optimality conditions are linear (only t h e derivative of the objective function appears), and the solution to a system of linear inequalities is of polynomial length in t h e size of t h e coefficients. So for quadratic problems an optimal continuous solution is a polynomial function of t h e length of t h e input. In addition, the proof of impossibility for strongly polynomial algorithms using t h e algebraic tree computation model, is not applicable to t h e quadratic case (although for t h e comparison computation model t h e proof is valid and it is impossible to derive strongly polynomial algorithms using only comparisons). . P P All this raises a n u m b e r of issues: are there strongly polynomial algorithms for quadratic separable convex network flow? Are there strongly polynomial algorithms for t h e nonseparable convex continuous case (where polynomial algorithms exist)?, and what classes of nonseparable quadratic integer problems, that are generally N P hard, are solvable in polynomial time?. We survey here some results related to these questions.

4.1

Strongly Polynomial Algorithms for the Separable Case

Using a known transformation, any m i n i m u m cost network flow problem can be written as a Transportation problem. It is therefore sufficient in t h e search for efficient algorithms for t h e quadratic separable network flow problem to focus on t h e quadratic separable Transportation problem. . P P T h e Q u a d r a t i c Transportation problem ( Q T P ) is defined on a bipartite network, with k supply nodes and n dem a n d nodes. T h e cost of transporting flow from a supply node to a d e m a n d node is a convex quadratic function of t h e flow quantity. T h e formulation of t h e continuous problem is as follows:

k

min

n

i

J2 ]CK'T'J + oMij]

Polynomial

&z Strongly

S. I .

Polynomial

y

Algorithms

&ij — $\

Optimization

81

' — -I-5 • * • } ™

5 > i i = dj xn > 0

for Convex

j = l,...,n i = l,...,fc,

(QTP)

j = l,...,n

where 6,-j > 0, s,- > 0, and dj > 0 are rational numbers and £],- 5,- = 22j 4 r It is not known whether t h e quadratic Transportation problem is solvable in strongly polynomial t i m e . W h e n t h e n u m b e r of supply nodes k is fixed, Cosares and Hochbaum gave a strongly polynomial algorithm. T h e algorithms presented in [CH90] exploit t h e relationship between t h e Transportation problem and t h e Allocation problem. T h e continuous Allocation problem can be solved by identifying a Lagrangean multiplier associated with t h e single constraint. In t h e q u a d r a t i c case this can be done in linear time. T h e algorithm for t h e Q u a d r a t i c Transportation problem entails relaxing and aggregating source constraints, and then searching for o p t i m a l values for t h e Lagrangean multipliers. For t h e case of two supply nodes, k = 2, t h e algorithm is linear. For greater values of k, t h e algorithm has running t i m e of 0(nk+1). A recent result by Megiddo and Tamir, [MT91], which invokes an alternative searching m e t h o d , yields a linear running t i m e of t h e algorithm for fixed k (the constant coefficient is exponential in k). T h e Allocation problem may be viewed as a single source Transportation problem. We use this problem to illustrate t h e technique of converting a continuous solution t o an integer solution. Allocation problems are all characterized by solvability using t h e greedy algorithm t h a t amounts to adding one unit increment to a variable if its marginal contribution to t h e objective function is minimized. T h e greedy algorithm however is not a polynomial algorithm. For t h e simple Allocation problem, m i n { ^ " = 1 fi(xi)\ E)" =1 Xi = B,x > 0 } , the running t i m e of t h e greedy is 0(B). T h e most general case of Allocation problem involves separable convex optimization over polymatroidal constraints: Given a submodular rank function r : 2E —* R, for E = {1, ..., n), i.e. r() = 0 and for all A,B C E, r(A) + r(B) > r{A U B) + r(A D B). T h e polymatroid defined by t h e rank function r, is t h e polytope {x\^2jeAXj r(A), A C E}. We call t h e system of inequalities {Y.jzAxi < r(^)> A C E}, polymatroidal constraints. T h e general Allocation problem, GAP, is (GAP)

min

£

/,•(*,•)

jEE

Y.

XJ

< r(A)

ACE

jeA

Xj > lj and integer j € E.

< the

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T h e network allocation problem is special case of (GAP) as proved in [FG86a] and [FG86b]. As such it is solvable in pseudo polynomial t i m e by t h e greedy algorithm. It is defined with respect t o any network (or graph), with a single source - root, and a set of sinks. Let G = (V, A) b e a directed network with node set V and arc set A. Let s G V be t h e source and T C V be t h e set of sinks. Let t h e total supply of t h e root, B > 0 be given. Let Cuv be t h e capacity limit on each arc (u,v). Let t h e vector of t h e flow b e = (<j>uv : (u,v) € A). min

£ cijXj + 0

£

4>vu-

(D,U)£A

ii)

£ £

su— £

4>uv = 0

v£V-T-{s}

^ 5} Siep 2: Let d = B8 - A

Step 3: If

If 5iep ^ : If

lid = d then S T O P , 5 = S'. lid>d t h e n 8\ H d < d then 8* then: Set I = {i e 7|a,- <S}, L = {a,-|i e 7} Set A = A - £ % £ = £ - T ^ , 5 <